A 4-choosable graph that is not (8:2)-choosable

Abstract

In 1980, Erdős, Rubin and Taylor asked whether for all positive integers a, b, and m, every (a : b)-choosable graph is also (am : bm)-choosable. We provide a negative answer by exhibiting a 4-choosable graph that is not (8 : 2)-choosable.Coloring the vertices of a graph with sets of colors (that is, each vertex is assigned a fixed-size subset of the colors such that adjacent vertices are assigned disjoint sets) is a fundamental notion, which in particular captures fractional colorings. The fractional chromatic number of a graph G can indeed be defined to be the infimum (which actually is a minimum) of the ratios a/b such that, if every vertex of G is replaced by a clique of order b and every edge of G is replaced by a complete bipartite graph between the relevant cliques, then the chromatic number of the obtained graph is at most a.In their seminal work on list coloring, Erdős, Rubin and Taylor [2] raised several intriguing questions about the list version of set coloring. Before stating them, let us review the relevant definitions. Set coloring.A function that assigns a set to each vertex of a graph is a set coloring if the sets assigned to adjacent vertices are disjoint. For positive integers a and b ≤ a, an (a : b)-coloring of a graph G is a set coloring with range {1,...,a} b , i.e., a set coloring that to each vertex of G assigns a b-element subset of {1, . . . , a}. The concept of (a : b)-coloring is a generalization of the conventional vertex coloring. In fact, an (a : 1)-coloring is exactly an ordinary proper a-coloring.A list assignment for a graph G is a function L that to each vertex v of G assigns a set L(v) of colors. A set coloring ϕ of G is an L-set coloring if ϕ(v) ⊆ L(v) for every v ∈ V (G). For a positive integer b, we say that ϕ is anFor an integer a ≥ b, we say that G is (a : b)-choosable if G is (L : b)-colorable for every list assignment L * This project falls within the scope of L.I.A. STRUCO.

Multiple list coloring of 3‐choice critical graphs

“…Assume the last sentence of this claim is not true, say c c′ j j is light for P 0 and P 1 but not light for P 2 . Note that by Claim 5, z 1 (2) ≥ , so if c c′ j j is heavy for P 2 , then there exists a distinct couple c c′ k k which is safe for P 2 . By the first part of this claim, c c′ k k is safe for neither P 0 nor P 1 .…”

“…It was conjectured by Erdős et al [1] that every a b ( , )choosable graph is am bm ( , )-choosable. This conjecture was refuted recently by Dvořák et al [2] who proved that for any integer k 4…”

“…Thus c c′ j j is not heavy for P 2 . Note that x 1 (2) ≥ , there exists a couple c c′ k k which is heavy for P 2 . It follows that c c c c…”

Multiple list colouring triangle free planar graphs

“…It was conjectured by Erdős, Rubin and Taylor [2] that if a graph G is (a, b)-choosable, then for any positive integer m, G is (am, bm)-choosable. If this conjecture were true, then ch * f (G) ≤ ch(G) for any graph G. However, very recently, this conjecture is refuted by Dvořák, Hu and Sereni [3] who proved that for any integer k ≥ 4, there is a k-choosable graph which is not (2k, 2)-choosable. Nevertheless, for a graph G with ch(G) = k, to show that ch * f (G) ≤ k, it suffices to show that for any integer m ≥ 1, G is (km + 1, m)choosable.…”

A characterization of (4,2)‐choosable graphs

“…1 , then G contains two vertex-disjoint cycles (one in ∪ P P 1 3 and one in P vw + 2 ), a contradiction. So, by symmetry between P 1 and P 2 , we assume ∈ x P Int( ) 3 1 and ∈ y P Int( )…”